Though I have included a few old puzzles that have interested the world for generations, where I felt that there was something new to be said about them, the problems are in the main original. It is true that some of these have become widely known through the press, and it is possible that the reader may be glad to know their source. On the question of Mathematical Puzzles in general there is, perhaps, little more to be said than I have written elsewhere. The history of the subject entails nothing short of the actual story of the beginnings and development of exact thinking in man.
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Though I have included a few old puzzles that have interested the world for generations, where I felt that there was something new to be said about them, the problems are in the main original. It is true that some of these have become widely known through the press, and it is possible that the reader may be glad to know their source. On the question of Mathematical Puzzles in general there is, perhaps, little more to be said than I have written elsewhere.
The history of the subject entails nothing short of the actual story of the beginnings and development of exact thinking in man. The historian must start from the time when man first succeeded in counting his ten fingers and in dividing an apple into two approximately equal parts. Every puzzle that is worthy of consideration can be referred to mathematics and logic.
Every man, woman, and child who tries to "reason out" the answer to the simplest puzzle is working, though not of necessity consciously, on mathematical lines. It is, in fact, not easy to say sometimes where the "empirical" begins and where it ends. When a man says, "I have never solved a puzzle in my life," it is difficult to know exactly what he means, for every intelligent individual is doing it every day.
If there were no puzzles to solve, there would be no questions to ask; and if there were no questions to be asked, what a world it would be! We should all be equally omniscient, and conversation would be useless and idle. Yet some of those examples that look the simplest should not be passed over without a little consideration, for now and again it will be found that there is some more or less subtle pitfall or trap into which the reader may be apt to fall.
It is good exercise to cultivate the habit of being very wary over the exact wording of a puzzle. It teaches exactitude and caution. But some of the problems are very hard nuts indeed, and not unworthy of the attention of the advanced mathematician. Readers will doubtless select according to their individual tastes. In many cases only the mere answers are given. This leaves the beginner something to do on his own behalf in working out the method of solution, and saves space that would be wasted from the point of view of the advanced student.
On the other hand, in particular cases where it seemed likely to interest, I have given rather extensive solutions and treated problems in a general manner. Where it is possible to say a thing in a manner that may be "understanded of the people" generally, I prefer to use this simple phraseology, and so engage the attention and interest of a larger public. The mathematician will in such cases have no difficulty in expressing the matter under consideration in terms of his familiar symbols.
I have taken the greatest care in reading the proofs, and trust that any errors that may have crept in are very few. If any such should occur, I can only plead, in the words of Horace, that "good Homer sometimes nods," or, as the bishop put it, "Not even the youngest curate in my diocese is infallible. Forsooth, a great arithmetician.
The puzzles in this department are roughly thrown together in classes for the convenience of the reader. Some are very easy, others quite difficult. Also, the arithmetical and algebraical puzzles are not separated in the manner adopted by some authors, who arbitrarily require certain problems to be solved by one method or the other.
The reader is left to make his own choice and determine which puzzles are capable of being solved by him on purely arithmetical lines. In every business of life we are occasionally perplexed by some chance question that for the moment staggers us. How long would it have taken you to think it out? The precocity of some youths is surprising. One is disposed to say on occasion, "That boy of yours is a genius, and he is certain to do great things when he grows up;" but past experience has taught us that he invariably becomes quite an ordinary citizen.
It is so often the case, on the contrary, that the dull boy becomes a great man. You never can tell. Nature loves to present to us these queer paradoxes. It is well known that those wonderful "lightning calculators," who now and again surprise the world by their feats, lose all their mysterious powers directly they are taught the elementary rules of arithmetic. A boy who was demolishing a choice banana was approached by a young friend, who, regarding him with envious eyes, asked, "How much did you pay for that banana, Fred?
Three countrymen met at a cattle market. It was found that five cobblers spent as much as four tailors; that twelve tailors spent as much as nine hatters; and that six hatters spent as much as eight glovers. The puzzle is to find out how much each of the four parties spent. The name of the particular game is of no consequence. They played seven games, and, strange to say, each won a game in turn, in the order in which their names are given. The puzzle is to find out how much money each man had with him before he sat down to play.
During how many years could the charity be administered? Of course, by "different ways" is meant a different number of men and women every time. He directed that every son should receive three times as much as a daughter, and that every daughter should have twice as much as their mother.
A charitable gentleman, on his way home one night, was appealed to by three needy persons in succession for assistance. To the first person he gave one penny more than half the money he had in his pocket; to the second person he gave twopence more than half the money he then had in his pocket; and to the third person he handed over threepence more than half of what he had left.
On entering his house he had only one penny in his pocket. Now, can you say exactly how much money that gentleman had on him when he started for home? A man recently bought two aeroplanes, but afterwards found that they would not answer the purpose for which he wanted them.
Did he make a profit on the whole transaction, or a loss? And how much? His family had been taking him around buying Christmas presents. I came out this morning with a certain amount of money in my pocket, and I find I have spent just half of it. In fact, if you will believe me, I take home just as many shillings as I had pounds, and half as many pounds as I had shillings. It is monstrous! Resting at noon within a tavern old, They all agreed to have a feast together.
So, for two shillings more than his due share Each honest man who had remained was bled. They settled later with those rogues, no doubt. How many were they when they first set out? It is a curious fact that there is only one other sum of money, in pounds, shillings, and pence all similarly repetitions of one figure , of which the digits shall add up the same as the digits of the amount in pence.
What is the other sum of money? Now, try to discover the smallest sum of money that can be written down under precisely the same conditions. It requires just a little judgment and thought. The multiplier must be regarded as an abstract number. It is true that two feet multiplied by two feet will make four square feet. Similarly, two pence multiplied by two pence will produce four square pence! And it will perplex the reader to say what a "square penny" is. But we will assume for the purposes of our puzzle that twopence multiplied by twopence is fourpence.
Now, what two amounts of money will produce the next smallest possible result, the same in both cases, when added or multiplied in this manner? The two amounts need not be alike, but they must be those that can be paid in current coins of the realm. Morgan G. Bloomgarten, the millionaire, known in the States as the Clam King, had, for his sins, more money than he knew what to do with.
It bored him. So he determined to persecute some of his poor but happy friends with it. They had never done him any harm, but he resolved to inoculate them with the "source of all evil. Another rule of his was that he would never give more than six persons exactly the same sum. Now, how was he to distribute the 1,, dollars?
You may distribute the money among as many people as you like, under the conditions given.
Amusements in Mathematics by Henry Ernest Dudeney
Biographies index Henry Ernest Dudeney came from a family which had a mathematical tradition and also a tradition of school teaching. Henry had one older brother Thomas born about He had four younger sisters Lucy born about , Kate born about , Emily born about , and Alice born about Henry learnt to play chess at a young age and soon became interested in chess problems.
Amusements in Mathematics
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Amusements in Mathematics
Amusements in Mathematics (PDF)